课程名称︰代数导论二 课程性质︰系必修 课程教师︰陈其诚 开课学院：理学院 开课系所︰数学系 考试日期（年月日）︰2012年3月22日 考试时限（分钟）：15:30~18:20 (170分钟) 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : Algebra 2 Exam 1 03/22 2012 Write your answer on the answer sheet. We give partial points. In this examination. R denotes a ring, F denotes a field, R[X](resp. R[X,Y]) denotes polynomial ringsover R in variable X(resp. X,Y). Let F_p denotes the finite field of order p. (1)(42 points) "yes" or "no". Either give a brief reason or give a counter example. 7 points each. (a) The ring Z[X] is a unique factorization ring. (b) Every ideal in Q[X,Y] is principle. (c) The polynomial 1+X+......+X^22 is irreducible in Q[X]. (d) The element √2+√3 is the root of a degree a polynomial in Q[X]. (e) The element √2+√3 is the root of a degree a polynomial in Q(√6)[X]. (f) If α€ Q(√2), then either Q(α)=Q or Q(α)=Q(√2). (2)(40 points) Prove the following assertions. 10 points each. (a) For each prime number p, there exists an irreducible quadratic polynom- ial in F_p[X]. (b) The ring Q[X]/(x^3+22) is actually a field. (c) The ring F_7[X]/(x^11-4) consists of exactly 7^11 elements. (d) If f(X)€ F[X] is of degree d, then in F there exist at most d roots of f(X). (3)(10 points) Let A be a finitely generated abelian group written additively. For each prime number p, let pA := {px | x€A}. Show that there exists a non-negative integer d such that for almost all p, the quotient group A/pA is isomorphic to A1⊕...⊕Ad, where each Ai is a cyclic group of order p (Hint: write A as athe direct sum of cyclic groups). (4)(8 points) Let M denote the direct sum of the Q[X]-module M1 = Q[X]/(x-1) and M2 = Q[X]/(x+1). Can M be generated by just one element? Prove or disprove it. --

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